Skip to main content
SpringerLink
Log in

Multidimensional Discrete Hilbert-Type Inequalities, Operators and Compositions

  • Chapter
  • First Online:
Analytic Number Theory, Approximation Theory, and Special Functions

Abstract

Hilbert-type inequalities with their operators are important in analysis and its applications. In this paper by using the methods of weight coefficients and technique of real analysis, a multidimensional discrete Hilbert-type inequality with a best possible constant factor is given. The equivalent forms, two types of reverses, a more accurate inequality with parameters, as well as a strengthened version of Hardy-Hilbert’s inequality with Euler constant are obtained. We also consider the relating operators with the norms, some particular examples and the compositions of two discrete Hilbert-type operators in certain conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alladi, K., Milovanovic, G.V., Rassias, M.Th. (eds.): Analytic Number Theory, Approximation Theory and Special Functions. Springer, New York (2013, to appear)

    Google Scholar 

  2. Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1984)

    Google Scholar 

  3. Arpad, B., Choonghong, O.: Best constant for certain multilinear integral operator. J. Inequal. Appl. 2006, Article ID 28582, 12 (2006)

    Google Scholar 

  4. Azar, L.: On some extensions of Hardy-Hilbert’s inequality and Applications. J. Inequal. Appl. 2009, Article ID 546829, 14 (2009)

    Google Scholar 

  5. Edwards, H.M.: Riemann’s Zeta Function. Dover Publications, New York (1974)

    MATH  Google Scholar 

  6. Erdos, P., Suranyi, J.: Topics in the Theory of Numbers. Springer, New York (2003)

    Book  Google Scholar 

  7. Gao, M.Z.: On an improvement of Hilbert’s inequality extended by Hardy-Riesz. J. Math. Res. Exposition 14(2), 255–259 (1994)

    MATH  MathSciNet  Google Scholar 

  8. Gao, M.Z., Yang, B.C.: On the extended Hilbert’s inequality. Proc. Am. Math. Soc. 126(3), 751–759 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hardy, G.H., Wright, E.W.: An Introduction to the Theory of Numbers, 5th edn. Clarendon Press, Oxford (1979)

    MATH  Google Scholar 

  10. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1934)

    Google Scholar 

  11. Hong, Y.: On Hardy-Hilbert integral inequalities with some parameters. J. Inequal. Pure Appl. Math.6(4), Art 92, 1–10 (2005)

    Google Scholar 

  12. Iwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Rhode Island (2004)

    Google Scholar 

  13. Krnić, M., Pečarić, J.E.: Hilbert’s inequalities and their reverses. Publ. Math. Debrecen 67(3–4), 315–331 (2005)

    MATH  MathSciNet  Google Scholar 

  14. Kuang, J.C.: Introduction to Real Analysis. Hunan Education Press, Chansha, China (1996)

    Google Scholar 

  15. Kuang, J.C.: Applied Inequalities. Shangdong Science Technic Press, Jinan, China (2004)

    Google Scholar 

  16. Kuang, J.C., Debnath, L.: On Hilbert’s type inequalities on the weighted Orlicz spaces. Pacific J. Appl. Math. 1(1), 95–103 (2007)

    MATH  MathSciNet  Google Scholar 

  17. Landau, E.: Elementary Number Theory, 2nd edn. Chelsea, New York (1966)

    Google Scholar 

  18. Li, Y.J., He, B.: On inequalities of Hilbert’s type. Bull. Aust. Math. Soc. 76(1), 1–13 (2007)

    Article  MATH  Google Scholar 

  19. Miller, S.J., Takloo-Bighash, R.: An Invitation to Modern Number Theory. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  20. Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer, Boston (1991)

    Book  MATH  Google Scholar 

  21. Pan, Y.L., Wang, H.T., Wang, F.T.: On Complex Functions. Science Press, Beijing (2006)

    Google Scholar 

  22. Yang, B.C.: On Hilbert’s integral inequality. J. Math. Anal. Appl. 220, 778–785 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  23. Yang, B.C.: A mixed Hilbert-type inequality with a best constant factor. Int. J. Pure Appl. Math. 20(3), 319–328 (2005)

    MATH  MathSciNet  Google Scholar 

  24. Yang, B.C.: Hilbert-Type Integral Inequalities. Bentham Science Publishers Ltd., Sharjah (2009)

    Google Scholar 

  25. Yang, B.C.: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing (2009)

    Google Scholar 

  26. Yang, B.C.: A half-discrete Hilbert-type inequality. J. Guangdong Univ. Educ. 31(3), 1–7 (2011)

    Google Scholar 

  27. Yang, B.C.: Discrete Hilbert-Type Inequalities. Bentham Science Publishers Ltd., Sharjah (2011)

    Google Scholar 

  28. Yang, B.C.: A half-discrete Hilbert-type inequality with a non-homogeneous kernel and two variables. Mediterr. J. Math. (2012). doi:10.1007/s00009- 012- 0213-50 (online first)

    Google Scholar 

  29. Yang, B.C.: Hilbert-type integral operators: norms and inequalities. In: Paralos, P.M., et al. (eds.) Nonlinear Analysis, Stability, Approximation, and Inequalities, pp. 771–859. Springer, New York (2012)

    Google Scholar 

  30. Yang, B.C.: Two Types of Multiple Half-Discrete Hilbert-Type Inequalities. Lambert Academic Publishing, Saarbrüken (2012)

    Google Scholar 

  31. Yang, B.C., Chen, Q.: A half-discrete Hilbert-type inequality with a homogeneous kernel and an extension. J. Inequal. Appl. 124 (2011). doi:10.1186/1029-242X-2011-124

    Google Scholar 

  32. Yang, B.C., Debnath, L.: On new strengthened Hardy-Hilbert’s inequality. Int. J. Math. Math. Soc. 21(2), 403–408 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  33. Yang, B.C., Gao, M.Z.: On a best value of Hardy-Hilbert’s inequality. Adv. Math. 26(2), 159–164 (1997)

    MATH  MathSciNet  Google Scholar 

  34. Yang, B.C., Krnić, M.: On the norm of a multi-dimensional Hilbert-type operator. Sarajevo J. Math. 7(20), 223–243 (2011)

    MathSciNet  Google Scholar 

  35. Yang, B.C., Rassias, Th.M.: On the way of weight coefficient and research for Hilbert-type inequalities. Math. Inequal. Appl. 6(4), 625–658 (2003)

    MATH  MathSciNet  Google Scholar 

  36. Yang, B.C., Rassias, Th.M.: On a Hilbert-type integral inequality in the subinterval and its operator expression. Banach J. Math. Anal. 4(2), 100–110 (2010)

    MATH  MathSciNet  Google Scholar 

  37. Yang, B.C., Brnetić, I., Krnić, M., Pečarić, J.E.: Generalization of Hilbert and Hardy-Hilbert integral inequalities. Math. Inequal. Appl. 8(2), 259–272 (2005)

    MATH  MathSciNet  Google Scholar 

  38. Zhao, D.J.: On a refinement of Hilbert double series theorem. Math. Practices Theory, 23(1), 85–90 (1993)

    Google Scholar 

  39. Zhong, W.Y.: The Hilbert-type integral inequality with a homogeneous kernel of Lambda-degree. J. Inequal. Appl. 2008, Article ID 917392, 13 (2008)

    Google Scholar 

  40. Zhong, W.Y.: A mixed Hilbert-type inequality and its equivalent forms. J. Guangdong Univ. Educ. 31(5), 18–22 (2011)

    MATH  Google Scholar 

  41. Zhong, W.Y.: A half discrete Hilbert-type inequality and its equivalent forms. J. Guangdong Univ. Educ. 32(5), 8–12 (2012)

    MATH  Google Scholar 

  42. Zhong, J.H.: Two classes of half-discrete reverse Hilbert-type inequalities with a non-homogeneous kernel. J. Guangdong Univ. Educ. 32(5), 11–20 (2012)

    Google Scholar 

  43. Zhong, W.Y., Yang, B.C.: A best extension of Hilbert inequality involving several parameters. J. Jinan Univ. (Nat. Sci.) 28(1), 20–23 (2007)

    Google Scholar 

  44. Zhong, W.Y., Yang, B.C.: On multiple Hardy-Hilbert’s integral inequality with kernel. J. Inequal. Appl. 2007, Article ID 27962, 17 (2007). doi:10.1155/ 2007/27

    Google Scholar 

  45. Zhong, W.Y., Yang, B.C.: A reverse Hilbert’s type integral inequality with some parameters and the equivalent forms. Pure Appl. Math. 24(2), 401–407 (2008)

    MathSciNet  Google Scholar 

  46. Zhong, J.H., Yang, B.C.: On an extension of a more accurate Hilbert-type inequality. J. Zhejiang Univ. (Sci. Ed.) 35(2), 121–124 (2008)

    Google Scholar 

Download references

Acknowledgements

This work is supported by 2012 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2012KJCX0079).

Author information

Authors and Affiliations

  1. Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong, 510303, P. R. China

    Bicheng Yang

Authors
  1. Bicheng Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bicheng Yang .

Editor information

Editors and Affiliations

  1. Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia

    Gradimir V. Milovanović

  2. Department of Mathematics, ETH Zürich, Zürich, Switzerland

    Michael Th. Rassias

Additional information

Dedicated to Professor Hari M. Srivastava

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Yang, B. (2014). Multidimensional Discrete Hilbert-Type Inequalities, Operators and Compositions. In: Milovanović, G., Rassias, M. (eds) Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0258-3_15

Download citation

Publish with us

Policies and ethics

Search

Navigation